Strain gauges · Measurement

Tensile/compressive force on a rod from measured strain

Work back from the longitudinal strain measured on a tension or compression member – or from the bridge signal in mV/V – to the axial stress and the force actually acting; plus the sensitivity of the measuring point as a calibration figure.

DMSKeil 2017
01

Inputs

Uniform tension/compression over the section at the gauge location, no notches or load-introduction points nearby (Keil recommends distance from the load introduction), no superimposed torsion. Bending is rejected only by the full bridges with opposite longitudinal gauges. Linear-elastic; temperature response and lead resistance not included.

02

Results

Enter values and run the calculation.

Method

What is calculated?

A slender member in tension or compression has its first principal direction along its axis, so a longitudinally bonded gauge directly measures the strain from which Hooke's law gives the axial stress. Multiplied by the cross-section area this yields the force. Keil presents exactly this relation in sec. 10.2.1 for the support rods of a roller-press bearing: 10 kN on an 8 mm rod with E = 208 000 N/mm² give 956 µm/m – run backwards here, 956 µm/m give 10 kN again. The full bridge of two longitudinal and two transverse gauges produces 1.2949 mV/V per 10 kN by eq. (10.5).

Equations

ε_l = 4·(U_M/U_B)/(k·B) (Keil Gl. 4.8 bzw. 10.1/10.2, nur Modus Signal)

σ = E·ε_l (Hookesches Gesetz, Keil Gl. 9.24 für einachsigen Zug)

A = π·d²/4 bzw. b·h

F = σ·A (Keil Abschn. 10.2.1: A = F/σ umgestellt)

Empfindlichkeit = (U_M/U_B)/F = k·B·ε_l/(4·F) (Keil Gl. 10.5)

Limits

Assumptions and typical mistake

Uniform tension/compression over the section at the gauge location, no notches or load-introduction points nearby (Keil recommends distance from the load introduction), no superimposed torsion. Bending is rejected only by the full bridges with opposite longitudinal gauges. Linear-elastic; temperature response and lead resistance not included.

With a single gauge, interpreting the value of a bending-loaded side as pure axial force; choosing the wrong bridge circuit in signal mode (error up to a factor 2.6); using the section of a different location than the gauge position (e.g. thread core instead of shank).

Inputs

What you enter – and where the values come from

What is your measured value?
Choose which number you have. If the amplifier in strain-gauge mode already shows a strain in µm/m (micrometres per metre; 1000 µm/m = 0.1 % change of length), choose strain. If it only shows the raw Wheatstone-bridge signal in mV/V (millivolts of output per volt of excitation), choose signal; gauge factor and bridge circuit are then needed for the conversion.
Measured longitudinal strain ε_l [µm/m]
The strain shown by a gauge bonded along the rod. Positive means the rod gets longer (tension), negative shorter (compression). For a full bridge this is the strain of one single longitudinal gauge, not the sum of all four. Source: the amplifier reading after zero balancing with the rod unloaded.
Bridge signal U_M/U_B [mV/V]
The raw bridge signal: Wheatstone-bridge output divided by the excitation, in mV/V. Keil's support rod gives about 1.29 mV/V at 10 kN. Source: the amplifier reading in mV/V mode after zero balancing.
Gauge factor k
The gauge factor is the sensitivity of the strain gauge: it states how much the electrical resistance changes when the gauge is stretched (ΔR/R = k·ε). Without it no strain can be computed from an electrical signal. Source: printed on every gauge package or in the manufacturer datasheet, typically 2.0 to 2.1 for constantan foil gauges; valid at room temperature. The same value must be set in the amplifier.
Bridge circuit
How the gauges are wired – this sets the bridge factor B, i.e. how many times the longitudinal strain is contained in the signal. One gauge: B = 1. One longitudinal and one transverse gauge: the transverse gauge measures the contraction −ν·ε and raises the signal to B = 1+ν. Two opposite longitudinal gauges plus two compensation gauges: B = 2, bending cancels. Two longitudinal and two transverse (Keil's measuring element): B = 2(1+ν) ≈ 2.58, bending- and temperature-compensated. Source: your own wiring or the element's datasheet.
Cross-section
How the load-bearing area at the gauge location is computed: circle from the diameter, rectangle from width times thickness, or you enter the area directly (e.g. from a drawing, for a tube or a profile). Only the section exactly where the gauges are bonded counts.
Diameter d [mm]
The rod diameter at the gauge location, circle only. The area grows with the square – a 1 % error in diameter is a 2 % error in force. Source: calliper; for threaded rods the shank diameter at the gauge location, not the thread core.
Width b [mm]
The width of the rectangular section at the gauge location (rectangle only). For flat bars the dimension the gauge is bonded on. Source: calliper or drawing.
Thickness h [mm]
The thickness of the rectangular section at the gauge location (rectangle only), i.e. the dimension perpendicular to the bonding surface. Source: calliper or drawing.
Cross-section area A [mm²]
The cross-section area directly in mm², only when ‘enter area directly’ is selected. Useful for tubes (π/4·(d_o²−d_i²)), profiles or values from the drawing. Keil's example: 8 mm round → 50.3 mm².
Young's modulus E [N/mm²]
Young's modulus describes how stiff the part's material is – how much stress it takes to produce a given strain (σ = E·ε). It is needed to turn measured or computed strains into stresses and vice versa. Source: material tables or datasheet; steel ≈ 210 000 N/mm², aluminium ≈ 70 000 N/mm², titanium ≈ 110 000 N/mm². E drops at elevated temperature.
Poisson's ratio ν
Poisson's ratio states how much a material contracts transversely when stretched longitudinally: ε_transverse = −ν·ε_longitudinal. It matters because transversely bonded gauges measure exactly this contraction and because both directions interact in biaxial stress states. Source: material tables; steel 0.28–0.30, aluminium 0.33, plastics 0.35–0.45.
Context

What this calculator is for

Force measurement on tie rods, support struts, anchors, bolt shanks and cables with a measuring body; checking preloads; design and calibration of simple rod-type force-measuring elements.

What comes next

  1. Install the gauges exactly along the rod – for a full bridge two of them opposite each other so bending cancels – and balance the bridge with the rod unloaded.
  2. Under load read the strain or the signal, choose the mode and enter the value.
  3. Enter the section at the gauge location and the material data; compare F with the expected or allowable force and use the sensitivity as the calibration value for the amplifier.
Results

How to read the results

  • The calculator works backwards: from the measured strain (or the signal) via the stress to the force. The stress is at the same time the value for the strength check, the force the value for the load assumption.
  • If the force differs clearly from expectation, check first: correct section at the gauge location, correct bridge circuit, no bending component (with a single gauge, measure once on the opposite side).
  • Force per 1 µm/m and sensitivity are the calibration figures of the measuring point – they stay the same as long as section, material and wiring remain unchanged.
Result quantities

What each value means

Longitudinal strain used [µm/m]
The longitudinal strain of one single gauge used in the calculation – in strain mode your entered value, in signal mode the strain converted back from mV/V, gauge factor and bridge factor. Lets you check the bridge circuit was chosen correctly.
Axial stress σ [N/mm²]
The tensile or compressive stress in the rod, i.e. force per area. Compare with the material's yield strength or allowable stress; for steel, 200 N/mm² is about half the yield strength of S355.
Cross-section area A [mm²]
The load-bearing area computed from your section choice or taken over directly. Check: 8 mm round → 50.27 mm².
Axial force F [kN]
The axial force the rod actually transmits at the gauge location – the calculator's real answer. Positive = tension, negative = compression. Compare with the expected load or the allowable force.
Force per 1 µm/m [N]
The stiffness of the measuring point: this much force corresponds to 1 µm/m of strain (E·A). Lets you convert further readings from the same measuring point in your head.
Sensitivity of the measuring point [mV/V je kN]
The bridge signal per kilonewton of force. Enter this as the sensitivity in the amplifier so it reads directly in kN. Keil's element: 0.129 mV/V per kN. Best confirmed by a calibration with a known force.
Source

Technical basis

Keil, Dehnungsmessstreifen, 2nd ed. 2017, sec. 10.2.1 axially loaded objects, eqs. (10.1)–(10.5), Tab. 10.1; sec. 9.4, eq. (9.24); sec. 4.2, eq. (4.8).

The source supports the equation structure and worked examples; this calculator does not replace calibration of the measuring chain.

FAQ

Frequently asked questions

Why does a single gauge sometimes read too high or too low under pure tension?

Because almost every rod is slightly bent as well – on one side the bending strain adds, on the other it subtracts. Two opposite longitudinal gauges in the bridge average this out (Keil sec. 10.2.1).

What does the sensitivity in mV/V per kN mean?

This much bridge signal is produced by 1 kN of force. It is used to set the amplifier so it reads directly in kN; Keil recommends confirming it by calibration in a testing machine.

Does the calculator apply to compressive forces too?

Yes, the strain is then entered negative and the force comes out negative. Slender compression members can buckle, though – that is a stability question, not a stress question.